One class of problems that biophysicists (and biologists, engineers,
and mathematicians) have thought about a lot is
called scaling
laws by physicists and
called allometric
laws by biologists: how does some interesting or important
feature y of a living organism such as its life span,
heart beat, breathing rate, metabolic power, surface area, length of
DNA, number of cell types, maximum speed, rate of reproduction,
or brain
size depend on some basic biological parameter x such
as the organism's mass M or maximum length L?
You might think that there would not be much to say since you might
expect arbitrary functional dependences y=f(x) that have no
common or simple form. But it is a remarkable and non-obvious
empirical fact that many biological relations have a simple
approximate power-law form y = c xα called a
scaling law, where c is some positive constant and
where α is called the "scaling exponent". Scientists would
then like to understand why and when such simple laws hold, and to
understand theoretically the values of the constants c and
α.
One of the first people to think and write about scaling laws was Galileo Galilei, who around 1630 deduced a simple scaling law to explain why large animals like elephants did not and could not look like greatly magnified small animals like ants. (These same arguments explain why movie monsters like Godzilla and King Kong could not exist as shown in the movies, the legs of these animals would break as soon as they tried to stand.)
The above figure, taken from the paper How fast do living organisms move: maximum speeds from bacteria to elephants and whales by Nicole Meyer-Vernet and Jean-Pierre Rospars (American Journal of Physics 83(8):719-722 (2015)), is a good example of what an empirical scaling law looks like and an excellent example of how scientists can develop a satisfying and useful insight about why the scaling law holds.
The horizontal axis of this figure is the total body mass in kg of organisms ranging from a tiny bacterium on the far left to the largest animals (elephants and whales) on the far right. Note how this axis spans an enormous dynamic range of 1022. The vertical axis is the maximum speed Vmax of the animal (which itself varies over a range of about 107) divided by its largest lateral length L (which also varies over a range of about 107), i.e. the vertical axis is the maximum speed per length. The data includes 202 running species (157 mammals plotted in magenta and 45 non-mammals plotted in green), 127 swimming species, and 91 micro-organisms. (The red and blue asterisks for M ≅ 102 kg are respectively the human world records for running and swimming, not very impressive I am afraid compared to other animals of comparable mass.) This remarkable plot has a simple punchline: the maximum speed of an animal divided by its length is approximately independent (within a factor of 10) of the animal's mass across all swimming and running species. The approximately constant value of this ratio is 10 in SI units, i.e., nearly all running and swimming organisms have a maximum speed of about ten body lengths per second. (As explained in the paper, flying animals don't follow this pattern and so are not included in the plot.)
The authors of this paper show that this approximate universal law, Vmax/L ≅ 10 s-1, can be deduced from the fact that all animals generate forces at the molecular level using nearly identical molecular motors. Bacteria use one or a few of these motors while the muscles of big animals like a human use huge numbers of similar molecular motors in parallel (which explains why the force exerted by a muscle is proportional to the cross-sectional area of the muscle). Using the fact that a single molecular motor has a known maximum applied force per cross-sectional area, and that the same motor has a maximum metabolic rate per mass, the authors derive a theoretical relation which is the horizontal black line in the figure that passes rather nicely through the middle of the data. So the approximately constant maximum speed per unit length boils down to the fact that all animals use a universal molecular motor, a very neat insight. (And this result turn raises another question: are the molecular motors in all of these different animals so similar because they have all evolved to attain some not-yet-known physical limit?)
Another important insight from the figure is that there is much scatter of the data about the expected theoretical result (horizontal black line). While one can obtain useful quantitative biophysics insights about biological phenomena, the comparison of theory with experimental data can be much noisier and so more challenging in biophysics than in other areas of physics like condensed matter or particle physics. There is much biological variation of features in living organisms and a challenge in biophysics is how to detect and describe quantitative patterns that may be hidden by the variations.